A Fluid Limit for the Engset Model

An Application to Retrial Queues

Stylianos Georgiadis

, Pascal Moyal

, Tam´as B´erczes

and J´anos Sztrik

Laboratoire de Math´ematiques Appliqu´ees de Compi`egne, Universit´e de Technologie de Compi`egne,

Centre de Recherches de Royallieu, BP 20529, 60205, Compi`egne, France

Faculty of Informatics, University of Debrecen, Egyetem tr 1, P.O. Box 12, 4010 Debrecen, Hungary

Keywords:

Engset Model, Fluid Limit, Semi-martingale Decomposition, Retrial Queues.

Abstract:

We represent the classical Engset-loss model by the stochastic process counting the number of customers in

the system. A ﬂuid limit for this process is established for all the possible values of the various parameters

of the system, as the number of servers tends to inﬁnity along with the number of sources. Our results are

derived through a semi-martingale decomposition method. A numerical application is provided to illustrate

these results. Then, we represent a ﬁnite-source retrial queue considering in addition the number of sources in

orbit. Finally, we extend the ﬂuid limit results to a retrial queueing system, discussing different cases.

1 INTRODUCTION

In many real-life queueing systems of ﬁnite capacity,

a customer may ﬁnd a full system upon arrival. In sev-

eral ﬁnite-source models, this requestcan return to the

source and stay there for a randomly distributed time

until it tries again to reach a server. The Engset model

represents a loss queueing system having this input

mechanism for several ﬁnite sources producing Pois-

son processes of the same intensity (see, e.g., (Engset,

1918)). We suppose that the system has no buffer,

hence a request is either immediately served or im-

mediately lost, whenever no server is available upon

arrival.

Such a model has been applied to a variety of re-

alistic computer and telecommunication systems and

networks. For exemple, an Engset system is adequate

to represent a radio-mobilenetwork in which the radio

sources emit messages only if no message of the same

source is currently in service. One could think that the

radio sources re-emit the same message as long as the

latter is refused due to the fact that all channels are

busy, and wait to re-issue a new message whenever

the previous message is in treatment.

This model has a wide ﬁeld of applications, so it

has been studied extensively through analytical and

algorithmic methods as well. However, when the sys-

tem becomes very large, several complexity problems

may appear. The ﬂuid limit technique offers the possi-

bility to approximate the exact values of some charac-

teristics of the system, when one or more parameters

tend to inﬁnity. In our case, the number of servers

tends to inﬁnity along with the number of sources.

Such techniques have been applied fruitfully to many

queueing systems (Robert, 2000; Asmussen, 2003;

Anisimov, 2007; Decreusefond and Moyal, 2012).

Recently, (Feuillet and Robert, 2012) constructed ex-

ponential martingales for the Engset model, allow-

ing to derive asymptotic estimates for several hitting

times of interest. We build on these results to de-

rive the ﬂuid limit of an Engset model having a single

server (Section 3), and then several servers (Section

4). Simulations are presented in Section 5.

In a ﬁnite source retrial queue, the messages

which could not reach a serverare sent to the so-called

orbit, from which they are re-emitted on and on, at a

rate that is possibly higher than the original one. It is

then easily seen that the Engset model in nothing but a

particular case of a retrial queueing system for which

the two emission rates are equal. Based on this obser-

vation, in Section 6 we investigate some applications

of our initial result to derive the ﬂuid approximation

of a retrial queue, under various conditions on the sys-

tem parameters.

2 THE ENGSET MODEL

We consider an Engset system with S (S ≥ 1) servers.

There are K (K > S) independent Poisson sources

emitting requests with intensity λ. The service times

315

Georgiadis S., Moyal P., Bérczes T. and Sztrik J..

A Fluid Limit for the Engset Model - An Application to Retrial Queues.

DOI: 10.5220/0004203101170122

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 117-122

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

of the requests are exponentially distributed of param-

eter µ. Whenever a request ﬁnds all servers busy, it is

immediately lost. If not, the request enters service

and the corresponding source remain inactive during

its treatment. As soon as the latter service has been

completed, the source becomes active again, and re-

emit jobs according to a Poisson process of intensity λ

that is independent from the past. In particular, there

is no dependence between the holding times and idle

periods of the sources.

Let X

:= (X

(t); t ≥ 0) denote the process count-

ing the number of customers in the system (i.e., the

number of busy servers) at current time. X

is a

Markov process, whose stationary measure is well-

known and easily derived.

We ﬁrst examine the Engset queueing model with

a single server, and denote X := X

the corresponding

process.

2.1 Semi-martingale Decompositions

Recall (Jacod and Shiryaev, 2003; Decreusefond and

Moyal, 2012) that for a givenFeller Markovprocess Z

of state space E and inﬁnitesimal generator A deﬁned

for all bounded f : E → R by

A f(i) = lim

h→0



E[ f (X(h)) | X(0) = i]− f(i)



, i∈E,

the process

: t → f (Z(t)) − f (Z(0)) −

A f (Z(s)) ds (1)

is a martingale w.r.t. the natural ﬁltration of Z.

For n ≥ 1 ﬁxed, consider an Engset system

M/M/n/n/nK, and add a superscript

to all the param-

eters involved. It is easily seen that the inﬁnitesimal

generator A

of X

reads for all bounded f : R → R

and all i ∈ {0, ..., n},

f(i) =











λ(K −i)( f(i+ 1) − f(i)) + µi( f(i−1) − f(i)),

i ∈{1, ..., n−1};

µn( f(n−1) − f(n)), i = n.

So, taking f as the identity function of {0, ..., n} in

(1), we get

f(i) = λ(nK −i)1

{i<n}

−iµ,

which leads to the following semi-martingale decom-

position :

(t) = X

(0) −µ

(s)ds

+ λ

(nK −X

(s))1

(s)<n}

ds+ M

(t),

where M

is a martingale and X

(0) ∈ [0, n].

2.2 The Free Process

The free process describes an Engset model without

limitation in the number of servers, hence an inﬁnite

server queues M/M/∞/∞/nK, having the same input

mechanism. As above, the process Y

counting the

number of customers in the system, satisﬁes the semi-

martingale decomposition

(t) = Y

(0) −µ

(s)ds

+ λ

(nK −Y

(s)) ds+ P

(t)

= Y

(0) −(λ + µ)

(s)ds+ λnKt + P

(t),

(2)

where P

is a martingale.

3 FLUID LIMIT

We are interested in the asymptotic behavior of X

(properly rescaled), as the number of servers goes to

inﬁnity together with the number of sources. We ob-

tain hereafter a ﬂuid limit that coincides with that of a

loss Erlang system M/M/n/n, and proceed as in Sec-

tion 6.7 of (Robert, 2000).

We normalize the various processes as follows.

For all t ≥ 0,

(t) =

(t)

;

(t) =

(t)

Assume that the deterministic initial condition satis-

ﬁes

(0)−→

n→∞

where x ∈ [0, 1] ﬁxed.

We easily check that the semi-martingale equation

(2) is similar to that of an M/M/∞ system of arrival in-

tensity λnK and service durations of parameter λ+ µ.

Whenever

(0)−→

n→∞

x, it then follows from Theorem

6.13 of (Robert, 2000) that for all T ≥ 0,



sup

0≤t≤T

(t) −Y

∗

(t) |



−→

n→∞

0, (3)

where, for all t ≥ 0,

∗

(t) = α+ (x−α)e

−(λ+µ)t

, (4)

setting

α =

λK

λ+ µ

Let the hitting times

:= inf{t ≥ 0;

(t) = 1}

= inf{t ≥ 0; X

(t) = n}

= inf{t ≥ 0; Y

(t) = n} (5)

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316

and

τ = inf{t ≥ 0;Y

∗

(t) = 1}. (6)

3.1 Heavy Trafﬁc

First, we examine the case α > 1. Then, we get

τ =

λ+ µ

log



α−x

α−1



. (7)

The following lemma follows from Proposition 6 of

(Feuillet and Robert, 2012).

Lemma 1. In heavy trafﬁc, the following convergence

in probability holds for the hitting time (5) :

−→

n→∞

τ, (8)

where τ is deﬁned by (7).

Theorem 1. For all ε > 0 and all T ≥ 0, we have



sup

0≤t≤T

(t) −X

∗

(t) |> ε



−→

n→∞

where, for all t ≥ 0,

∗

(t) = 1∧



α+ (x−α)e

−(λ+µ)t



Proof. We have



sup

0≤t≤T

(t) −X

∗

(t) |> ε



≤ P



sup

0≤t≤T∧τ

(t) −X

∗

(t) |> ε



+ P



inf

≤t≤T

(t) < 1 −



+ P



sup

≤t≤T

| 1−X

∗

(t) |>



. (9)

By the continuity of X

∗

(.), the ﬁrst and third term on

the r.h.s. of (9) vanish, respectively in view of (3)

and (8). The second term vanishes as it is less than

≤t), where

is the hitting time of ⌊

nε

⌋+ 1 by

, the congestion process of an M/M/1 queue with

arrival intensity nµ and service rate λn(K −1). As

nµ < λn(K −1), this queue is stable, so it is a classical

result that

is of the order of



λ(K−1)



⌊

nε

⌋+1

3.2 Light Trafﬁc

Assume now that α < 1.

Theorem 2. In light trafﬁc, for all ε > 0 and all T ≥

0, we have



sup

0≤t≤T

(t) −X

∗

(t) |> ε



−→

n→∞

where in that case, for all t ≥ 0,

∗

(t) = Y

∗

(t) = α+ (x−α)e

−(λ+µ)t

. (10)

Proof. Notice that



sup

0≤t≤T

(t) −X

∗

(t) |



≤ P(τ

≤ T) + P



sup

0≤t≤T

(t) −Y

∗

(t) |> ε



and apply (3) together with Markov inequality. The

ﬁrst term vanishes as τ

is asymptotically of the or-

der of (nα

)

−1

, as can be shown along the lines of

(Feuillet and Robert, 2012).

3.3 Critical Case

Suppose that α = 1. We reason as above :

Theorem 3. In the critical case, τ

is of the order of

log

√

n, hence the same convergence as in Theorem 2

holds true.

4 MULTISERVER ENGSET

MODEL

We now check that the results of Section 3 still hold

true for an Engset queue with S identical servers and

the process X

counting the busy servers. We consider

the Engset model M/M/nS/nS/nK. Easily, we obtain

the inﬁnitesimal generator A

of X

f(i) = λ(nK −i)1

{i<nS}

−iµ,

and the corresponding semi-martingale decomposi-

tion

(t) = X

(0) −µ

(s)ds

+ λ

(nK −X

(s))1

(s)<nS}

ds+ M

(t),

where M

is a martingale and X

(0) ∈ [0, nS]. The

free process Y

is given by

(t) = Y

(0) −(λ + µ)

(s)ds+ λnKt + P

(t),

where P

is a martingale. For all t ≥ 0, we consider

the normalized processes

(t) and

(t) over n and

the limit of the initial condition :

(0)−→

n→∞

where x

∈[0, S] ﬁxed. Moreover, we observe that (3)

holds in the case of S servers with Y

∗

(t) = Y

∗

(t) and

= inf{t ≥ 0; Y

(t) = nS},

= inf{t ≥ 0;Y

∗

(t) = S}.

Consequently, the hitting time τ

converges in proba-

bility to

λ+ µ

log



α−x

α−S



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317

Theorem 4. For all ε > 0 and all T ≥ 0, we have



sup

0≤t≤T

(t) −X

∗

(t) |> ε



−→

n→∞

where, for all t ≥ 0, we consider the following cases :

if α > S,

∗

(t) = S ∧



α+ (x

−α)e

−(λ+µ)t



; (11)

if α ≤ S,

∗

(t) = α+ (x

−α)e

−(λ+µ)t

. (12)

As a consequence, the ﬂuid limit for the Engset

system can be derived for any arbitrary, but ﬁxed, val-

ues of the number of sources and servers.

5 NUMERICAL RESULTS

In this section, we present a numerical example for

the multiserver Engset model concerningthe different

cases discussed in Section 4. Consider a M/M/S/S/K

queueing system with parameters S = 500, λ = 0.1

and µ = 0.5. The critical value for the number of

sources is, therefore, K = 3000. Setting various val-

ues for the number of sources, we may obtain the

heavy trafﬁc (α > 500), the light trafﬁc (α < 500) or

the critical case (α = 500). We set the initial value

= 100 for the busy servers.

In the following ﬁgures, we present a realization

of the process X

of the model described above in the

time interval [0, 50], along with the ﬂuid limit, for the

three cases as given in (11) and (12).

Notice that, in Figure 1, the hitting time of S

servers for the realization (4.6381) is fairly close to

the theoretical value of τ

(5.3648).

6 APPLICATION TO SINGLE

SERVER RETRIAL QUEUES

Retrial queues follow the following scenario : when a

customer arriveswith all servers and waiting positions

(if any) being busy, he leaves the service area but after

some randomly distributed time repeats his demand.

For a review of the main results on the topic see (Falin

and Templeton, 1997) and the references therein.

The general queueing system with retrials is de-

scribed more precisely as follows. There are ﬁnitely

many identical independent fully available servers at

which requests arrive. Each source can generate a re-

quest with rate λ. If an arriving request ﬁnds at least

one server free, it immediately occupies the server

0 5 10 15 20 25 30 35 40 45 50

100

150

200

250

300

350

400

450

500

Time

Busy Servers

Trajectory

Fluid Limit

Figure 1: Heavy trafﬁc (K > 3000).

0 5 10 15 20 25 30 35 40 45 50

100

150

200

250

300

350

400

450

500

Time

Busy Servers

Trajectory

Fluid Limit

Figure 2: Light trafﬁc (K < 3000).

0 5 10 15 20 25 30 35 40 45 50

100

150

200

250

300

350

400

450

500

Time

Busy Servers

Trajectory

Fluid Limit

Figure 3: Critical case (K = 3000).

and leaves it after completion of service. The rate of

service time is denoted by µ. If all servers are busy

at arrival time, then the source goes into the orbit (a

secondary queue of inﬁnite size) and starts the gener-

ation of requests with rate ν until it ﬁnds a free server.

After completion of service, the source returns to the

initial state and it can generate a new request, while

the server may serve a new request. All the times in-

volved in the model are assumed to be mutually inde-

pendent of each other.

The presence of the orbit makes the retrial queue-

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ing model more ﬂexible than the Engset one, once the

source may generate a message with a different rate

from the rate of the initial state. Actually, an Engset

queue is a special case of the certain ﬁnite-source re-

trial system under the condition ν = λ. Because of this

fact, we can apply the results of the previous section

to a retrial queue, and, therefore, investigate different

cases under diverse system parameters.

Consider a single-server retrial queue with ﬁnitely

many sources. Let E = {0, 1, . . . , K − 1} and F =

{0, 1}. Let N := (N(t); t ≥ 0) be the number of

sources of in orbit and C := (C(t); t ≥ 0) the num-

ber of busy servers, with state spaces E and F respec-

tively. The system state at time t can be described

by the coupled process A(t) := (N(t),C(t)). The pro-

cess A := (A(t); t ≥ 0) is a continuous-time Markov

process with ﬁnite state space E ×F. Since the state

space of the process A is ﬁnite, the process is ergodic

with stationary measure

π(·, ·) deﬁned as follows

π(i, j) = lim

t→∞

P(N(t) = i,C(t) = j), i ∈E, j ∈ F.

We consider a sequence of retrial systems with the n-

th one having n servers and nK sources. Let E

{0, 1, . . . , nK −1} and F

= {0, 1, . . . , n}. Let A

) be the corresponding process for the n-th

system, with state space E

×F

For all i ∈ E

, j ∈ F

, we consider the functions

f(i, j) := i and g(i, j) := j. Then, the inﬁnitesimal

operator Q

of A

applied to f and g reads

f(i, j) = −iν1

{j<n}

+ (nK −i−n)λ1

{j=n}

and

g(i, j) = [iν+ (nK −i− j)λ − jµ]1

{j<n}

−nµ1

{j=n}

which yields the following semi-martingale decom-

positions :

(t) =N

(0) −ν

(s)ds

+ (ν−λ)

(s)1

(s)=n}

+ λn(K −1)

(s)=n}

ds+ M

(t);

=ν

(s)1

(s)<n}

+ λ

(nK −N

(s) −n)1

(s)=n}

and

(t) =C

(0) + (ν −λ)

(s)1

(s)<n}

+ λ

(nK −C

(s))1

(s)<n}

−µ

(s)ds+ M

(t);

(nK −N

(s) −C

(s))1

{

(s)<n}

νN

(s)1

(s)<n}

+ nµ

(s)=n}

ds.

6.1 Law of Large Numbers

We apply the same normalization as in Section 3:

(t) =

(t)

;

(t) =

(t)

;

(t) =

(t)

;

(t) =

(t)

Assume that

(0)−→

n→∞

;

(0)−→

n→∞

where n

∈ [0, K] and c

∈ [0, 1]. Thus, for all t ≥ 0,

we obtain

∗

(t) =n

+ λ(K −1)

∗

(s)=1}

+ (ν−λ)

∗

(s)1

∗

(s)=1}

−ν

∗

(s)ds; (13)

∗

(t) =c

+ (ν−λ)

∗

(s)1

∗

(s)<1}

+ λ

(K −C

∗

(s))1

∗

(s)<1}

−µ

∗

(s)ds. (14)

Theorem 5. The following weak convergence holds :

(

) ⇒ (N

∗

where the deterministic functions N

∗

and C

∗

are the

unique solutions of (13) and (14), respectively.

Proof. We follow the classical steps for proving weak

convergence of processes. The increasing processes

i and h

i vanish uniformly on any compact

AFluidLimitfortheEngsetModel-AnApplicationtoRetrialQueues

319

interval as n goes large. Moreover, the Aldous-

Reboledo tightness criterion for semi-martingales

(see, e.g. (Joffe and Mtivier, 1986)) is easily met by

both

and

. Thus, the sequence



(·),

(·)



tight, and any subsequential limit (N

∗

(·), C

∗

(·)) reads

for all t ≥ 0, the equations (13) and (14). The weak

limit is then a solution of this system. The unique-

ness of the latter is easily checked by showing that

the underlying mapping is locally-Lipschitz continu-

ous.

6.2 Discussion

We discuss applications of the latter result in several

cases.

(i) If λ = ν, from (4) and (14), C

∗

coincides with the

ﬂuid limit Y

∗

of Section 3, in the various cases.

Setting, whenever τ is ﬁnite (i.e. in the heavy traf-

ﬁc case),

= n

−λτ

from the equation (13), we obtain that

∗

(t) =n

−λt

{t<τ}

+ (K −1+ [τ

−(K −1)]e

−λ(t−τ)

{t≥τ}

We check as well the intuitive result that asymp-

totically, the orbit is either full (heavy trafﬁc case)

or empty (light trafﬁc/ critical case).

(ii) Suppose now that Kλ ≤ λ + µ, and ﬁx the initial

condition n

= 0. Let

= inf{t ≥ 0; N

(t) > 0}.

Up to ρ

, no arrival occurs from the orbit, so the

value of ν is irrelevant. It is easily seen that

= τ

in distribution,

where τ

corresponds to the previous hitting time

for an Engset model. So, from the results of the

previous section in the light trafﬁc and critical

cases, a proof similar to that of Theorem 2 shows

that, for all t ≥ 0,

∗

(t) = 0; C

∗

(t) = Y

∗

(t),

where Y

∗

(t) is given in (10).

(iii) All the same, if ﬁnally λ < ν and Kλ > ν+ µ, we

show by stochastic comparison of Markov pro-

cesses that

≤

where τ

corresponds to a heavily loaded Engset

model of arrival rate λ. So, Theorem 1 entails that

for some 0 < ρ ≤τ, where τ is deﬁned by (6),

∗

(t) = 1, for all t ≥ ρ,

i.e. the server is busy all the time after ρ, at the

ﬂuid level.

7 CONCLUSIONS

We have derived the ﬂuid limit of an Engset queueing

system with several servers. After discussing the con-

nection between the Engset queue and retrial queue-

ing models, we present several ﬂuid limit results for

retrial queues. The generalization of the ﬂuid limit

for all possibles values of the parameters λ, µ and ν

of the retrial model, and the numerical conﬁrmation

of their accuracy is a challenging problem that is cur-

rently under investigation.

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Falin, G. and Templeton, J. (1997). Retrial Queues. Chap-

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